What is |(x^2+1)/2 |<3 ?

The solution to |(x^2+1)/2 |<3 is -sqrt(5)<x<sqrt(5)

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|(x^2+1)/2 |<3

\frac{x ^{2} + 1}{2} < 3

- 5 < x < 5

Interval Notation

(- 5, 5)

Decimal Notation

- 2.23606 \dots < x < 2.23606 \dots

Solution steps

\frac{x ^{2} + 1}{2} < 3

Apply absolute rule: If

∣ u ∣ < a, a > 0

then

- a < u < a

- 3 < \frac{x ^{2} + 1}{2} < 3

\frac{x ^{2} + 1}{2} > - 3 and \frac{x ^{2} + 1}{2} < 3

True for all x and - 5 < x < 5

Merge Overlapping Intervals

- 5 < x < 5

5 + 3 x = 2 (x + 3)

12 x^{2} + 25 x + 2

\frac{x - 2}{x + 3} < 0

\frac{1}{3} - \frac{2}{3}

2 x^{2} + 4 x y